Quasi-alternating surgeries

Abstract

In this article, we explore phenomena relating to quasi-alternating surgeries on knots, where a quasi-alternating surgery on a knot is a Dehn surgery yielding the double branched cover of a quasi-alternating link. Since the double branched cover of a quasi-alternating link is an L-space, quasi-alternating surgeries are special examples of L-space surgeries. We show that all SnapPy census L-space knots admit quasi-alternating surgeries except for the knots t09847 and o9_30634, neither of which have any quasi-alternating surgeries. In particular, this finishes Dunfield's classification of the L-space knots among all SnapPy census knots. In addition, we show that all asymmetric census L-space knots have exactly two quasi-alternating slopes and that these are consecutive integers. Similar behavior is observed for some of the Baker-Luecke asymmetric L-space knots. We also classify the quasi-alternating surgeries on torus knots and show that the set of formal L-space slopes is either empty or infinite This allows us to give examples of asymmetric formal L-spaces.